Idealization and factualization in science

Summary This paper considers the method of idealization and factualization as the main method of all advanced empirical sciences. The procedure is as follows. Some idealizing conditions are assumed: the vanishing of factors (p _i=0) which never vanish in the real world. An idealization law is formulated — a law which is exactly (non-vacuously) fulfilled only in an ideal model, not in any real system. Then the idealizing assumptions are abrogated one by one-it is a process of gradual factualization, of the transition to the factual laws which are fulfilled in real systems. These laws may be directly applied and tested by experience.     

Exploring the impact of phonological awareness, visual–spatial working memory, and preschool quantity–number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study

This longitudinal study explored the importance of kindergarten measures of phonological awareness, working memory, and quantity–number competencies (QNC) for predicting mathematical school achievement in third graders (mean age 8 years 8 months). It was found that the impact of phonological awareness and visual–spatial working memory, assessed at 5 years of age, was mediated by early QNC, which predicted math achievement in third grade. Importantly, and confirming our isolated number words hypothesis, phonological awareness had no impact on higher numerical competencies (i.e., when number words needed to be linked with quantities [QNC Level II and above]) but predicted basic numerical competencies (i.e., when number words were isolated from quantities [QNC Level I]), explaining the moderate relationship between early literacy development and the development of mathematical competencies.     

The relationships between quantity‐number competencies,working memory,and phonological awareness in 5‐ and 6‐year‐olds

In this study, the interdependencies among phonological awareness, verbal working memory components, and early numerical skills in children 1 year before school entry are addressed. Early numerical skills were conceptualized as quantity‐number competencies (QNC) at both basic (QNC Level 1) and advanced (QNC Level 2) levels. In a sample of 1,343 children aged 5 and 6, structural equation modelling provided support for the isolated number words hypothesis (Krajewski & Schneider, 2009, J. Exp. Child Psychol., 103, 516–531). This hypothesis claims that phonological awareness contributes to the acquisition of QNC Level 1, such as learning the number word sequence, but not of QNC Level 2, which requires the linkage of number words to quantities. In addition, phonological awareness relied on verbal working memory, especially with regard to the phonological loop, central executive, and episodic buffer. The results were congruent with the idea that phonological awareness mediates the impact of verbal working memory on QNCs. The relationships between verbal working memory, phonological awareness, and QNCs were comparable in monolingual and bilingual children.